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Theorem List for Metamath Proof Explorer - 30901-31000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlhpmat 30901 An element covered by the lattice unit, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.)

Theoremlhpmatb 30902 An element covered by the lattice unit, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.)

Theoremlhp2at0 30903 Join and meet with different atoms under co-atom . (Contributed by NM, 15-Jun-2013.)

Theoremlhp2atnle 30904 Inequality for 2 different atoms under co-atom . (Contributed by NM, 17-Jun-2013.)

Theoremlhp2atne 30905 Inequality for joins with 2 different atoms under co-atom . (Contributed by NM, 22-Jul-2013.)

Theoremlhp2at0nle 30906 Inequality for 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)

Theoremlhp2at0ne 30907 Inequality for joins with 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)

Theoremlhpelim 30908 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 30901 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)

Theoremlhpmod2i2 30909 Modular law for hyperplanes analogous to atmod2i2 30733 for atoms. (Contributed by NM, 9-Feb-2013.)

Theoremlhpmod6i1 30910 Modular law for hyperplanes analogous to complement of atmod2i1 30732 for atoms. (Contributed by NM, 1-Jun-2013.)

Theoremlhprelat3N 30911* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 30283. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)

Theoremcdlemb2 30912* Given two atoms not under the fiducial (reference) co-atom , there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 30-May-2012.)

Theoremlhple 30913 Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)

Theoremlhpat 30914 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)

Theoremlhpat4N 30915 Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)

Theoremlhpat2 30916 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.)

Theoremlhpat3 30917 There is only one atom under both and co-atom . (Contributed by NM, 21-Nov-2012.)

Theorem4atexlemk 30918 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemw 30919 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempw 30920 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemp 30921 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemq 30922 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlems 30923 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemt 30924 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemutvt 30925 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempnq 30926 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemnslpq 30927 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemkl 30928 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemkc 30929 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemwb 30930 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempsb 30931 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemqtb 30932 Lemma for 4atexlem7 30946. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlempns 30933 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemswapqr 30934 Lemma for 4atexlem7 30946. Swap and , so that theorems involving can be reused for . Note that must be expanded because it involves . (Contributed by NM, 25-Nov-2012.)

Theorem4atexlemu 30935 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemv 30936 Lemma for 4atexlem7 30946. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemunv 30937 Lemma for 4atexlem7 30946. (Contributed by NM, 21-Nov-2012.)

Theorem4atexlemtlw 30938 Lemma for 4atexlem7 30946. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemntlpq 30939 Lemma for 4atexlem7 30946. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemc 30940 Lemma for 4atexlem7 30946. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemnclw 30941 Lemma for 4atexlem7 30946. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemex2 30942* Lemma for 4atexlem7 30946. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)

Theorem4atexlemcnd 30943 Lemma for 4atexlem7 30946. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemex4 30944* Lemma for 4atexlem7 30946. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)

Theorem4atexlemex6 30945* Lemma for 4atexlem7 30946. (Contributed by NM, 25-Nov-2012.)

Theorem4atexlem7 30946* Whenever there are at least 4 atoms under (specifically, , , , and ), there are also at least 4 atoms under . This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 30215, our is a shorter way to express . With a longer proof, the condition could be eliminated (see 4atex 30947), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)

Theorem4atex 30947* Whenever there are at least 4 atoms under (specifically, , , , and ), there are also at least 4 atoms under . This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 30215, our is a shorter way to express . (Contributed by NM, 27-May-2013.)

Theorem4atex2 30948* More general version of 4atex 30947 for a line not necessarily connected to . (Contributed by NM, 27-May-2013.)

Theorem4atex2-0aOLDN 30949* Same as 4atex2 30948 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex2-0bOLDN 30950* Same as 4atex2 30948 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex2-0cOLDN 30951* Same as 4atex2 30948 except that and are zero. TODO: do we need this one or 4atex2-0aOLDN 30949 or 4atex2-0bOLDN 30950? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex3 30952* More general version of 4atex 30947 for a line not necessarily connected to . (Contributed by NM, 29-May-2013.)

Theoremlautset 30953* The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)

Theoremislaut 30954* The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012.)

Theoremlautle 30955 Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlaut1o 30956 A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)

Theoremlaut11 30957 One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcl 30958 A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.)

TheoremlautcnvclN 30959 Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)

Theoremlautcnvle 30960 Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)

Theoremlautcnv 30961 The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Theoremlautlt 30962 Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcvr 30963 Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautj 30964 Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)

Theoremlautm 30965 Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlauteq 30966* A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)

Theoremidlaut 30967 The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)

Theoremlautco 30968 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

TheorempautsetN 30969* The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

TheoremispautN 30970* The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

Syntaxcldil 30971 Extend class notation with set of all lattice dilations.

Syntaxcltrn 30972 Extend class notation with set of all lattice translations.

SyntaxcdilN 30973 Extend class notation with set of all dilations.

SyntaxctrnN 30974 Extend class notation with set of all translations.

Definitiondf-ldil 30975* Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Definitiondf-ltrn 30976* Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Definitiondf-dilN 30977* Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)

Definitiondf-trnN 30978* Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.)

Theoremldilfset 30979* The mapping from fiducial co-atom to its set of lattice dilations. (Contributed by NM, 11-May-2012.)

Theoremldilset 30980* The set of lattice dilations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)

Theoremisldil 30981* The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Theoremldillaut 30982 A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)

Theoremldil1o 30983 A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)

Theoremldilval 30984 Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)

Theoremidldil 30985 The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)

Theoremldilcnv 30986 The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)

Theoremldilco 30987 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

Theoremltrnfset 30988* The set of all lattice translations for a lattice . (Contributed by NM, 11-May-2012.)

Theoremltrnset 30989* The set of lattice translations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)

Theoremisltrn 30990* The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Theoremisltrn2N 30991* The predicate "is a lattice translation". Version of isltrn 30990 that considers only different and . TODO: Can this eliminate some separate proofs for the case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)

Theoremltrnu 30992 Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom . Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.)

Theoremltrnldil 30993 A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)

Theoremltrnlaut 30994 A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremltrn1o 30995 A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.)

Theoremltrncl 30996 Closure of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrn11 30997 One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrncnvnid 30998 If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.)

TheoremltrncoidN 30999 Two translations are equal if the composition of one with the converse of the other is the zero translation. This is an analog of vector subtraction. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremltrnle 31000 Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012.)

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